Subgroup ($H$) information
| Description: | $C_9:C_3$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$bd^{8}, c^{2}d^{6}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $(C_3^2\times A_4):S_3$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9:C_3^2:S_4$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:S_3$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_3^2:S_3$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $C_3^2:S_3$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_3$ | ||
| Normalizer: | $C_3^3:S_3$ | ||
| Normal closure: | $C_3^2.A_4$ | ||
| Core: | $C_3^2$ | ||
| Minimal over-subgroups: | $C_3^2.A_4$ | $C_3\wr C_3$ | $C_9:C_6$ |
| Maximal under-subgroups: | $C_3^2$ | $C_9$ |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $-3$ |
| Projective image | $(C_3^2\times A_4):S_3$ |